Research

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13 pages, 305 KiB

Open AccessArticle

by Youssef Achtoun, Milanka Gardasević-Filipović, Slobodanka Mitrović and Stojan Radenović

Symmetry 2024, 16(4), 415; https://doi.org/10.3390/sym16040415 - 02 Apr 2024

Viewed by 424

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known [...] Read more.

This paper is dedicated to the memory of the esteemed Serbian mathematician Slaviša B. Prešić (1933–2008). The primary aim of this survey paper is to compile articles on Prešić-type mappings published since 1965. Additionally, it introduces a novel class of symmetric contractions known as Prešić–Menger and Prešić–Ćirić–Menger contractions, thereby enriching the literature on Prešić-type mappings. The paper endeavors to furnish young researchers with a comprehensive resource in functional and nonlinear analysis. The relevance of Prešić’s method, which generalizes Banach’s theorem from 1922, remains significant in metric fixed point theory, as evidenced by recent publications. The overview article addresses the growing importance of Prešić’s approach, coupled with new ideas, reflecting the ongoing advancements in the field. Additionally, the paper establishes the existence and uniqueness of fixed points in Menger spaces, contributing to the filling of gaps in the existing literature on Prešić’s works while providing valuable insights into this specialized domain. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

16 pages, 673 KiB

Open AccessArticle

Recent Developments in Iterative Algorithms for Digital Metrics

by Aasma Shaheen, Afshan Batool, Amjad Ali, Hamed Al Sulami and Aftab Hussain

Symmetry 2024, 16(3), 368; https://doi.org/10.3390/sym16030368 - 18 Mar 2024

Viewed by 896

Abstract

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, [...] Read more.

This paper aims to provide a comprehensive analysis of the advancements made in understanding Iterative Fixed-Point Schemes, which builds upon the concept of digital contraction mappings. Additionally, we introduce the notion of an Iterative Fixed-Point Schemes in digital metric spaces. In this study, we extend the idea of Iteration process Mann, Ishikawa, Agarwal, and Thakur based on the ϝ-Stable Iterative Scheme in digital metric space. We also design some fractal images, which frame the compression of Fixed-Point Iterative Schemes and contractive mappings. Furthermore, we present a concrete example that exemplifies the motivation behind our investigations. Moreover, we provide an application of the proposed Fractal image and Sierpinski triangle that compress the works by storing images as a collection of digital contractions, which addresses the issue of storing images with less storage memory in this paper. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

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14 pages, 288 KiB

Open AccessArticle

Application to Activation Functions through Fixed-Circle Problems with Symmetric Contractions

by Rizwan Anjum, Mujahid Abbas, Hira Safdar, Muhammad Din, Mi Zhou and Stojan Radenović

Symmetry 2024, 16(1), 69; https://doi.org/10.3390/sym16010069 - 04 Jan 2024

Cited by 1 | Viewed by 774

Abstract

In this paper, our main aim is to present innovative fixed-point theorems that provide solutions to the fixed-circle problem with symmetric contractions. We accomplish this by employing operator enrichment techniques within the context of Banach spaces. Furthermore, we demonstrate the practical application of [...] Read more.

In this paper, our main aim is to present innovative fixed-point theorems that provide solutions to the fixed-circle problem with symmetric contractions. We accomplish this by employing operator enrichment techniques within the context of Banach spaces. Furthermore, we demonstrate the practical application of these theorems by showcasing their relevance to the rectified linear unit (ReLU) activation function. By exploring the connection between fixed points and activation functions, our work contributes to a deeper understanding of the behavior and properties of these fundamental mathematical concepts. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

17 pages, 338 KiB

Open AccessArticle

On (α,p)-Cyclic Contractions and Related Fixed Point Theorems

by Victory Asem, Yumnam Mahendra Singh, Mohammad Saeed Khan and Salvatore Sessa

Symmetry 2023, 15(10), 1826; https://doi.org/10.3390/sym15101826 - 26 Sep 2023

Viewed by 855

Abstract

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as [...] Read more.

Lipschitz mapping appears inevitably in many branches of mathematics, especially in functional analysis, and leads to the study of new results in metric fixed point theory. Goebel and Sims (resp. Goebel and Japon-Pineda) introduced a class of the Lipschitz mappings termed as

(α, p)

-Liptschitz mappings and studied not only the modified form of the Lipschitz condition, but also the behavior of a finite number of their iterates. The purpose of this paper is to discuss the various types of

(α, p)

-contractions with cyclic representation that extend the results due to Banach, Kannan, and Chatterjea. Moreover, based on such types of contractions and the property of symmetry, we obtain some related fixed-point results in the setting of metric spaces. Some examples are studied to illustrate the validity of our obtained results. As an application of our results, we establish the existence of the solution to a class of Fredholm integral equations. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

29 pages, 358 KiB

Open AccessArticle

Geraghty–Pata–Suzuki-Type Proximal Contractions and Related Coincidence Best Proximity Point Results

by Naeem Saleem, Maneesha Tur Raazzia, Nawab Hussain and Asim Asiri

Symmetry 2023, 15(8), 1572; https://doi.org/10.3390/sym15081572 - 11 Aug 2023

Cited by 1 | Viewed by 527

Abstract

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context [...] Read more.

The objective of this research paper is to establish the existence and uniqueness of the best proximity and coincidence with best proximity point results, specifically focusing on Geraghty–Pata–Suzuki-type proximal mappings. To achieve this, we introduce three types of mappings, all within the context of a complete metric space: an

α - θ -

Geraghty–Pata–Suzuki-type proximal contraction; an

α - θ -

generalized Geraghty–Pata–Suzuki-type proximal contraction; and an

α - θ -

modified Geraghty–Pata–Suzuki-type proximal contraction. These new results generalize, extend, and unify various results from the existing literature. Symmetry plays a crucial role in solving nonlinear problems in operator theory, and the variables involved in the metric space are symmetric. Several illustrative examples are provided to showcase the superiority of our results over existing approaches. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

14 pages, 310 KiB

Open AccessArticle

Best Proximity Point Results for n-Cyclic and Regular-n-Noncyclic Fisher Quasi-Contractions in Metric Spaces

by Kamal Fallahi, Morteza Ayobian and Ghasem Soleimani Rad

Symmetry 2023, 15(7), 1469; https://doi.org/10.3390/sym15071469 - 24 Jul 2023

Cited by 1 | Viewed by 749

Abstract

In this work, we introduce some new concepts such as n-cyclic Fisher quasi-contraction mappings, full-n-noncyclic and regular-n-noncyclic Fisher quasi-contraction mappings in metric spaces. We then generalize the results by Safari-Hafshejani, Amini-Harandi and Fakhar. Meanwhile, we answer the question [...] Read more.

In this work, we introduce some new concepts such as n-cyclic Fisher quasi-contraction mappings, full-n-noncyclic and regular-n-noncyclic Fisher quasi-contraction mappings in metric spaces. We then generalize the results by Safari-Hafshejani, Amini-Harandi and Fakhar. Meanwhile, we answer the question “under what conditions does a full-n-noncyclic Fisher quasi-contraction mapping have

n (n - 1) / 2

unique optimal pairs of fixed points?”. Further, to support the main results, we highlight all of the new concepts via non-trivial examples. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

16 pages, 427 KiB

Open AccessArticle

Bipolar b-Metric Spaces in Graph Setting and Related Fixed Points

by Haroon Ahmad, Mudasir Younis and Afrah Ahmad Noman Abdou

Symmetry 2023, 15(6), 1227; https://doi.org/10.3390/sym15061227 - 08 Jun 2023

Cited by 1 | Viewed by 1023

Abstract

In this article, we develop a new notion that combines fixed-point theory and graph theory: graphical bipolar b-metric spaces. We demonstrate fixed-point solutions in the framework of graphical bipolar b-metric spaces, employing covariant and contravariant mapping contractions, which is a new [...] Read more.

In this article, we develop a new notion that combines fixed-point theory and graph theory: graphical bipolar b-metric spaces. We demonstrate fixed-point solutions in the framework of graphical bipolar b-metric spaces, employing covariant and contravariant mapping contractions, which is a new addition to this end. This article also features illustrative examples drawn from various contexts to further demonstrate our findings. This is a significant study since it melds ideas from graph theory with those from generalized bipolar metric spaces, and considers that the symmetry of the edges of the underlying graphs connected with the enunciated metric spaces is essential in the graphical metric spaces. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

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16 pages, 811 KiB

Open AccessArticle

Fractional Differential Boundary Value Equation Utilizing the Convex Interpolation for Symmetry of Variables

by Aftab Hussain

Symmetry 2023, 15(6), 1189; https://doi.org/10.3390/sym15061189 - 02 Jun 2023

Viewed by 839

Abstract

In this paper, we introduce a novel form of interpolative convex contraction and develop some new theorems by utilizing the progressive method of interpolative convex contractions. We also obtain some fixed point results for a Suzuki convex contraction in orbitally S-complete F [...] Read more.

In this paper, we introduce a novel form of interpolative convex contraction and develop some new theorems by utilizing the progressive method of interpolative convex contractions. We also obtain some fixed point results for a Suzuki convex contraction in orbitally S-complete F-metric spaces. The second purpose of this research is to evaluate the effectiveness of the fixed point approach in solving fractional differential equations with boundary conditions. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

18 pages, 355 KiB

Open AccessArticle

Feng-Liu’s Approach to Fixed Point Results of Intuitionistic Fuzzy Set-Valued Maps

by Mohammed Shehu Shagari, Trad Alotaibi, Rehana Tabassum, Awad A. Bakery, OM Kalthum S. K. Mohamed and Arafa O. Mustafa

Symmetry 2023, 15(4), 930; https://doi.org/10.3390/sym15040930 - 18 Apr 2023

Viewed by 920

Abstract

The applications of non-zero self distance function have recently been discovered in both symmetric and asymmetric spaces. With respect to invariant point results, the available literature reveals that the idea has only been examined for crisp mappings in either symmetric or asymmetric spaces. [...] Read more.

The applications of non-zero self distance function have recently been discovered in both symmetric and asymmetric spaces. With respect to invariant point results, the available literature reveals that the idea has only been examined for crisp mappings in either symmetric or asymmetric spaces. Hence, the aim of this paper is to introduce the notion of invariant points for non-crisp set-valued mappings in metric-like spaces. To this effect, the technique of

κ

-contraction and Feng-Liu’s approach are combined to establish new versions of intuitionistic fuzzy functional equations. One of the distinguishing ideas of this article is the study of fixed point theorems of intuitionistic fuzzy set-valued mappings without using the conventional Pompeiu–Hausdorff metric. Some of our obtained results are applied to examine their analogues in ordered metric-like spaces endowed with an order and binary relation as well as invariant point results of crisp set-valued mappings. By using a comparative example, it is observed that a few important corresponding notions in the existing literature are complemented, unified and generalized. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

Review

Jump to: Research

15 pages, 360 KiB

Open AccessReview

Solving the Hydrodynamical System of Equations of Inhomogeneous Fluid Flows with Thermal Diffusion: A Review

by Sergey V. Ershkov, Evgeniy Yu. Prosviryakov, Natalya V. Burmasheva and Victor Christianto

Symmetry 2023, 15(10), 1825; https://doi.org/10.3390/sym15101825 - 26 Sep 2023

Cited by 2 | Viewed by 843

Abstract

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown [...] Read more.

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment. Full article

(This article belongs to the Special Issue New Trends in Fixed Point Theory with Emphasis on Symmetry)

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